A318278 -id:A318278 - cp's OEIS frontend

A309141 Nonunitary highly composite numbers: numbers with a record number of nonunitary divisors.

1, 4, 8, 16, 24, 36, 48, 72, 144, 216, 288, 360, 576, 720, 1080, 1440, 2160, 2880, 3600, 4320, 5040, 7200, 7560, 10080, 15120, 20160, 25200, 30240, 45360, 50400, 60480, 75600, 100800, 110880, 151200, 221760, 277200, 302400, 332640, 453600, 498960, 554400, 665280

Offset: 1

Amiram Eldar, Jul 14 2019

nonn

Numbers k with A048105(k) > A048105(j) for all j < k.

The corresponding values of records are 0, 1, 2, 3, 4, 5, 6, 8, 11, 12, 14, 16, 17, 22, 24, 28, 32, 34, 37, 40, 44, 46, 48, 56, 64, 68, 74, 80, 84, 92, 96, ... (see the link for more values)

Amiram Eldar, Table of n, a(n) for n = 1..559
Amiram Eldar, Table of n, a(n), A048105(a(n)) for n = 1..559

Cf. A048105, A002182 (highly composite), A002110 (unitary), A037992 (infinitary), A293185 (bi-unitary), A318278 (exponential).

f[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; fm=-1; s={}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^5}]; s

A306736 Exponential infinitary highly composite numbers: where the number of exponential infinitary divisors (A307848) increases to record.

Original entry on oeis.org

1, 4, 36, 576, 14400, 705600, 57153600, 6915585600, 1168733966400, 337764116289600, 121932845980545600, 64502475523708622400, 40314047202317889000000, 33904113697149344649000000, 32581853262960520207689000000, 44604557116992952164326241000000, 74980260513665152588232411121000000

Offset: 1

Amiram Eldar, May 01 2019

nonn

Subsequence of A025487.
All the terms have prime factors with multiplicities which are infinitary highly composite number (A037992) > 1, similarly to exponential highly composite numbers (A318278) whose prime factors have multiplicities which are highly composite numbers (A002182). Thus all the terms are squares. Their square roots are 1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 18378360, 349188840, 8031343320, 200783583000, 5822723907000, 180504441117000, ...
Differs from A307845 (exponential unitary highly composite numbers) from n >= 107. a(107) = 2^24 * (3 * 5 * ... * 19)^6 * (23 * 29 * ... * 509)^2 ~ 2.370804... * 10^456, while A307845(107) = (2 * 3 * 5 * ... * 19)^6 * (23 * 29 * ... * 521)^2 ~ 2.454885... * 10^456.

Amiram Eldar, Table of n, a(n) for n = 1..201

Cf. A002182, A025487, A037445, A037992, A307845, A307848, A318278.

di[1] = 1; di[n_] := Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[All, 2]]]; fun[p_, e_] := di[e]; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); s = {}; am = 0; Do[a1 = a[n]; If[a1 > am, am = a1; AppendTo[s, n]], {n, 1, 10^6}]; s (* after Jean-François Alcover at A037445 *)

A307848(a(n)) = 2^(n-1).

A307845 Exponential unitary highly composite numbers: where the number of exponential unitary divisors (A278908) increases to a record.

Original entry on oeis.org

1, 4, 36, 576, 14400, 705600, 57153600, 6915585600, 1168733966400, 337764116289600, 121932845980545600, 64502475523708622400, 40314047202317889000000, 33904113697149344649000000, 32581853262960520207689000000, 44604557116992952164326241000000, 74980260513665152588232411121000000

Offset: 1

Amiram Eldar, May 01 2019

nonn

Subsequence of A025487.
All the terms have prime factors with multiplicities which are primorials > 1 (the primorials, A002110, are the unitary highly composite numbers), similarly to exponential highly composite numbers (A318278) whose prime factors have multiplicities which are highly composite numbers (A002182). Thus all the terms are squares. Their square roots are 1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 18378360, 349188840, 8031343320, 200783583000, 5822723907000, 180504441117000, ...
First differs from A306736 at n = 107. - Georg Fischer, Aug 13 2025

Amiram Eldar, Table of n, a(n) for n = 1..201

Cf. A002110, A002182, A025487, A278908, A306736, A318278.

f[p_, e_] := 2^PrimeNu[e]; a[n_] := Times @@ (f @@@ FactorInteger[n]); s = {}; am = 0; Do[a1 = a[n]; If[a1 > am, am = a1; AppendTo[s, n]], {n, 1, 10^6}]; s

A278908(a(n)) = 2^(n-1).

A335386 Tri-unitary highly composite numbers: where the number of tri-unitary divisors (A335385) increases to a record.

Original entry on oeis.org

1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 18378360, 349188840, 8031343320, 200783583000, 5822723907000, 180504441117000, 6678664321329000, 273825237174489000, 11774485198503027000, 553400804329642269000, 27116639412152471181000, 1437181888844080972593000

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..32

Analogous sequences: A002182 (highly composite), A002110 (unitary), A037992 (infinitary), A293185 (bi-unitary), A318278 (exponential), A306736 (exponential infinitary), A307845 (exponential unitary), A309141 (nonunitary), A322484 (semi-unitary).

Cf. A335385.

f[p_, e_] := If[e == 3 || e == 6, 4, 2]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); dm = 0; s = {}; Do[If[(d1 = d[n]) > dm, dm = d1; AppendTo[s, n]], {n, 1, 1100000}]; s

A335385(a(n)) = 2^(n-1).

A340233 a(n) is the least number with exactly n exponential divisors.

Original entry on oeis.org

1, 4, 16, 36, 65536, 144, 18446744073709551616, 576, 1296, 589824

Offset: 1

Amiram Eldar, Jan 01 2021

nonn

a(11) = 2^(2^10) has 309 digits and is too large to be included in the data section.

See the link for more values of this sequence.

			a(2) = 4 since 4 is the least number with 2 exponential divisors, 2 and 4.

Amiram Eldar, Table of n, a(n) for n = 1..12
Amiram Eldar, Table of n, a(n) for n = 1..100 (given by prime factorizations)

Subsequence of A025487.
Cf. A049419, A318278, A322791.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340232 (bi-unitary).

f[p_, e_] := DivisorSigma[0, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 6; s = Table[0, {max}]; c = 0; n = 1;  While[c < max, i = d[n]; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* ineffective for n > 6 *)

A049419(a(n)) = n and A049419(k) != n for all k < a(n).

A333931 Recursive highly composite numbers: numbers with a record number of recursive divisors (A282446).

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 420, 900, 1260, 4620, 6300, 13860, 44100, 55440, 69300, 180180, 485100, 720720, 900900, 2882880, 3063060, 6306300, 12252240, 15315300, 49008960, 58198140, 107207100, 232792560, 290990700, 931170240, 1163962800, 2036934900, 4655851200

Offset: 1

Amiram Eldar, Apr 10 2020

nonn

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 48, 54, ...

Amiram Eldar, Table of n, a(n) for n = 1..200

Subsequence of A025487.
Cf. A282446.
Analogous sequences: A002182 (highly composite), A002110 (unitary), A037992 (infinitary), A293185 (bi-unitary), A309141 (nonunitary), A318278 (exponential).

recDivNum[1] = 1; recDivNum[n_] := recDivNum[n] = Times @@ (1 + recDivNum/@ (Last /@ FactorInteger[n])); rm = 0; s = {}; Do[r = recDivNum[n]; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^4}]; s

A358253 Numbers with a record number of non-unitary square divisors.

Original entry on oeis.org

1, 8, 32, 128, 288, 864, 1152, 2592, 4608, 10368, 20736, 28800, 41472, 64800, 115200, 259200, 518400, 1036800, 2073600, 4147200, 8294400, 9331200, 12700800, 25401600, 50803200, 101606400, 203212800, 406425600, 457228800, 635040000, 812851200, 914457600, 1270080000

Offset: 1

Amiram Eldar, Nov 05 2022

nonn

Numbers m such that A056626(m) > A056626(k) for all k < m.

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 20, 22, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..283
Amiram Eldar, Table of n, a(n), A056626(a(n)) for n = 1..283

Cf. A056626, A358252.
Subsequence of A025487.
Similar sequences: A002182 (all divisors), A002110 (unitary), A037992 (infinitary), A046952 (square divisors), A053624 (odd divisors), A293185 (bi-unitary), A309141 (non-unitary), A318278 (exponential).

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; s = {}; fmax = -1; Do[If[(fn = f[n]) > fmax, fmax = fn; AppendTo[s, n]], {n, 1, 10^5}]; s

s(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + floor(f[i,2]/2)) - 2^sum(i = 1, #f~, 1 - f[i,2]%2);}
lista(nmax) = {my(smax = -1, sn); for(n = 1, nmax, sn = s(n); if(sn > smax, smax = sn; print1(n, ", "))); }

A365681 Numbers with a record number of exponentially squarefree divisors.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 60, 120, 180, 360, 840, 1260, 2520, 6300, 7560, 12600, 27720, 69300, 83160, 138600, 332640, 360360, 900900, 1081080, 1801800, 4324320, 5405400, 12612600, 17297280, 18378360, 30630600, 73513440, 86486400, 91891800, 214414200, 294053760

Offset: 1

Amiram Eldar, Sep 15 2023

nonn

Indices of records of A365680.

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..255
Amiram Eldar, Table of n, a(n), A365680(a(n)) for n = 1..255

Cf. A365680.
Subsequence of A025487.
Similar sequences: A306736, A307845, A318278.

f[p_, e_] := Count[Range[e], ?SquareFreeQ] + 1; d[1] = 1; d[n] := Times @@ f @@@ FactorInteger[n]; v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

A348342 Noninfinitary highly composite numbers: where the number of noninfinitary divisors (A348341) increases to a record.

Original entry on oeis.org

1, 4, 12, 16, 36, 48, 144, 240, 576, 720, 1680, 2880, 3600, 5040, 11520, 14400, 15120, 20160, 25200, 45360, 55440, 80640, 100800, 166320, 176400, 226800, 277200, 498960, 720720, 887040, 1108800, 1587600, 1940400, 2494800, 3603600, 6486480, 9979200, 11531520, 14414400

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

The record numbers of noninfinitary divisors are 0, 1, 2, 3, 5, 6, 11, 12, 13, 22, 24, 26, 37, 44, 46, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..468
Amiram Eldar, Table of n, a(n), A348341(a(n)) for n = 1..468

Cf. A348341.
Subsequence of A025487.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386.

nid[1] = 0; nid[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; dm = -1; s = {}; Do[If[(d = nid[n]) > dm, dm = d; AppendTo[s, n]], {n, 1, 10^6}]; s

A348632 Nonexponential highly composite numbers: where the number of nonexponential divisors (A160097) increases to a record.

Original entry on oeis.org

1, 6, 12, 24, 30, 60, 120, 210, 240, 360, 420, 720, 840, 1260, 1680, 2520, 3360, 5040, 7560, 9240, 10080, 15120, 18480, 25200, 27720, 36960, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 480480, 498960, 554400, 665280, 720720, 1081080, 1441440

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

The corresponding record values are 1, 3, 4, 6, 7, 10, 14, 15, 17, 20, 22, 24, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..651
Amiram Eldar, Table of n, a(n), A160097(a(n)) for n = 1..651

Cf. A160097.
Subsequence of A025487.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386.

f1[p_, e_] := e + 1; f2[p_, e_] := DivisorSigma[0, e]; ned[1] = 1; ned[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; dm = -1; s = {}; Do[If[(d = ned[n]) > dm, dm = d; AppendTo[s, n]], {n, 1, 10^6}]; s

cp's OEIS Frontend

A309141 Nonunitary highly composite numbers: numbers with a record number of nonunitary divisors.

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A306736 Exponential infinitary highly composite numbers: where the number of exponential infinitary divisors (A307848) increases to record.

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A307845 Exponential unitary highly composite numbers: where the number of exponential unitary divisors (A278908) increases to a record.

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A335386 Tri-unitary highly composite numbers: where the number of tri-unitary divisors (A335385) increases to a record.

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A340233 a(n) is the least number with exactly n exponential divisors.

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A333931 Recursive highly composite numbers: numbers with a record number of recursive divisors (A282446).

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A358253 Numbers with a record number of non-unitary square divisors.

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A365681 Numbers with a record number of exponentially squarefree divisors.

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A348342 Noninfinitary highly composite numbers: where the number of noninfinitary divisors (A348341) increases to a record.

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